Only math problems

For all problems, if an algorithm can verify an answer, can an algorithm find a solution?

prove all nontrivial zeros of the analytical continuation of the Riemann zeta function have a real part of 0.5

Does a set of all sets contain itself?

[math]\sqrt{-1}[/math]

Easy, but

Prove

[math] \displaystyle\displaystyle{\sup \bigcup_{k=1}^{n} X_k = \max(\sup X_1, \sup X_2, \cdots, \sup X_n) } [/math]

Does this extend to the infinite case?

[math] \displaystyle{\sup \bigcup_{k=1}^{\infty} X_k = ?} [/math]

>>9104521
yes

>>9104522
1i

>>9104519
not necessarily, so no.

>>9104521
>a set of all sets
You mean "THE set of all sets", right?
There can be only one.

Let [math]p[/math] be a prime number.
Prove that [math](p - 1)! + 1[/math] is divisible by [math]p[/math].

>>9104517
410

>>9104554
That's correct! The rule (rot13-ed):

Nqq gur ahzoref, gura chg gurve qvssrerapr va sebag bs vg.

http://www.rot13.com/

n+k=(n-k,n+k)

>>9104517
410
first digit is difference, subsequent are sum
>>9104522
>>9104534
i or -i
could also be 2 or 3 in the field with 5 elements
2*2 = 4 = -1 (mod 5)
3*3 = 9 = -1 (mod 5)
specify field pls
>>9104551
multiply all non-zero elements of F_p.
inverses cancel in pairs except for -1.
(p-1)! = -1 mod p

>>9104517
7+3=10
The other answers are wrong.

>>9104620
>>9104659
You shouldn't spoil the fun for others by writing up the solutions.

2^p-q^2=1999 solve in primes

>>9104733
[math]2^{11}-7^2=1999[/math]

Given a polynomial [math]P(x)\in\mathbb Z[x][/math], is there a way to find if there exist [math]x,\,y\in\mathbb N[/math] such that [math]P(x)=y^2[/math]?

>>9104921
how do you prove that there are no other solutions?

prove the following is false

x is a number in the multiplication table of 4 ⇒ x-1 is a prime number or x+1 is a a prime number

Are [math]\mathbb{Z}[i][/math] and [math]\mathbb{Z}[\sqrt{5}][/math] isomorphic as rings?

>>9105149
meant to say x is a product of 4, if anyone got confused by my wording

>>9105151
No, -1 has two squareroots in Z[i] but none in Z[sqrt(5)].

>>9105149
Let x=56 then neither x-1 nor x+1 are prime.

>>9105193
for a brainlet, can you explain why that's enough to say it's not isomorphic? I mean couldn't a homomorphism just map -1 to an element that has two square roots ?

>>9105293

Isomorphic implies elementarily equivalent.

Two structures M and N of the same signature σ are elementarily equivalent if every first-order sentence (formula without free variables) over σ is true in M if and only if it is true in N, i.e. if M and N have the same complete first-order theory..

To prove this, you use induction on the complexity of the formula.

>>9105293

f : Z[i] ---> Z[sqrt(5)]

[math]f(i)^2 = -1[/math]

Nothing in Z[sqrt(5)] satisfies x^2 = -1

>>9104527
Make it [math]\sup \left( \sup X_k \right)[/math] then it works for both.

>>9105293
The identity element of a ring is always preserved by nonzero homomorphisms of rings. It's easy to show additive inverses are as well.

>>9105312
based. thanks.

>>9105319
>It's easy to show additive inverses are as well.
oh damn you're right. thanks

>>9104517
410
ez

If you convert a base 10 prime into base 4 will the resulting number always be a prime in base 10?

>>9104519
Google "P = NP" or "P vs NP".

>>9104519
Through brute force, yes, but that's not efficient.

>>9105312
What about ZxZ? It should be isomorphic to Z[i]
>>9105319

>>9105442
Counterexample: 5

>>9105985
What multiplication do you take in ZxZ? If you use
(a,b) * (c,d) = (ac - bd, ad + bc)
then it's isomorphic to Z[i].

>>9106062
thanks mang.

/sci/ is actually pretty good some times

for example converting 5 into base 4 is 11. 11 is a prime in base 10. i've done this for about 3 days by hand to find if all primes converted to base 4 have a writen value that is a prime in base 10.
EX. (5,11,23,113,1301,110111,...)

>>9106021
[math]5_{10}=11_4[/math]
I'm pretty sure 11 is prime

>>9104520
suppose not. [math]\Rightarrow\Leftarrow[/math]
QED

>>9105442
[math]31_{10} = 133_{4}[/math]

133 = 7*19

>>9104562
uggcf://jjj.lbhghor.pbz/jngpu?i=u7WuxKtxEap